Finite Group Databasebeta

← #29665263

Subgroups — #29665263

Order 15616.

650 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455906C211yesmaximal, characteristic
2#21455823C211yesmaximal, characteristic, complemented
3#21454842C211yesmaximal, characteristic, complemented
4#21445253C212yesmaximal, complemented
5#21445225C212yesmaximal, complemented
6#21445215C212yesmaximal, complemented
7#21445191C212yesmaximal, complemented
8#21444599C211yesmaximal, characteristic, complemented
9#21444529C211yesmaximal, characteristic, complemented
10#21444481C211yesmaximal, characteristic, complemented
11C61×#3228C211yesmaximal, characteristic, complemented
12C2×C4×D244C2211yescharacteristic
13#16840090—21no—
14#16840078—21no—
15#16840049C2212yes—
16#16840049C2212yes—
17#16839964C2212yes—
18#16839954C2212yes—
19#16839912—21no—
20#16839895—21no—
21#16839872—22no—
22C4×#11758201—22no—
23C8×D244C2212yes—
24C8×D244C2212yes—
25#16839827—22no—
26#16839819C2211yescharacteristic
27#16839815C2211yescharacteristic
28#16839815C2211yescharacteristic
29C22⋊C8×D61—22no—
30#16839515C2212yescomplemented
31#16839513C2212yescomplemented
32#16839511C2212yescomplemented
33#16839511C2212yescomplemented
34#16839508C2212yescomplemented
35#16839506C2212yescomplemented
36C61×#533C2211yescharacteristic
37C61×#532C2211yescharacteristic
38C61×#500C2211yescharacteristic, complemented
39C61×#498C2211yescharacteristic, complemented
40C61×#486C2211yescharacteristic, complemented
41C61×#486C2211yescharacteristic, complemented
42C61×#482—22no—
43C61×#478—22no—
44C61×#431C2211yescharacteristic
45C22×D244D411yescharacteristic
46C22×C4×D61—21no—
47C2×#11758201—22no—
48C2×#11758199—22no—
49C2×C8×D61—22no—
50C2×C8×D61—22no—
51C4×D244—21no—
52C4×D244—21no—
53C4×D244D412yes—
54C4×D244C2312yes—
55C2×C4⋊Dic61D411yescharacteristic
56C2×#11758198—21no—
57#14519708—22no—
58#14519708—22no—
59C4⋊#92418C2312yes—
60C4⋊#92418C2312yes—
61C4⋊#92418C2312yes—
62#14519701—22no—
63#14519698—22no—
64#14519697—22no—
65#14519697—22no—
66C22⋊#92418—21no—
67C22⋊#92418—21no—
68C22⋊#92418—21no—
69#14519689—22no—
70#14519689—22no—
71#14519688—21no—
72C4×#92418—21no—
73C4×#92418—21no—
74C2×#41×C61—21no—
75C4⋊Q8×C61C2311yescharacteristic
76C4⋊Q8×C61C2311yescharacteristic
77C4⋊Q8×C61C2311yescharacteristic
78#127×C61C2311yescharacteristic
79#125×C61—21no—
80#124×C61—21no—
81C22⋊Q8×C61—21no—
82C22⋊Q8×C61—21no—
83#121×C61—21no—
84#121×C61—21no—
85C4×Q8×C61—21no—
86C4×Q8×C61—21no—
87C4×D4×C61—21no—
88C4×D4×C61—21no—
89#117×C61—21no—
90C2×C42×C61C2311yescharacteristic
91#107×C61—22no—
92#106×C61—22no—
93C4⋊C8×C61D412yes—
94#102×C61—22no—
95#102×C61—22no—
96C22⋊C8×C61D412yes—
97C4×C8×C61C2311yescharacteristic
98C4×C8×C61C2311yescharacteristic
99C23×D61—21no—
100#41×C61—82no—
101C2×Q8×C61—21no—
102C2×Q8×C61—21no—
103C2×Q8×C61—21no—
104C2×D4×C61—21no—
105C2×D4×C61—21no—
106C2×D4×C61—21no—
107C2×D4×C61—41no—
108C22×C4×C61—41no—
109C22×C4×C61C2×D411yescharacteristic
110C22×C4×C61C2×D411yescharacteristic
111C22×C4×C61#4111yescharacteristic
112C22×Dic61—21no—
113C2×D244—21no—
114C2×D244—24no—
115C2×D244C2×D412yes—
116C2×C4×D61—22no—
117C2×C4×D61—22no—
118C2×C4×D61—82no—
119C2×C8×C61C2×D412yes—
120C2×C8×C61C2×D412yes—
121C4⋊C4×C61—21no—
122C4⋊C4×C61—21no—
123C4⋊C4×C61—21no—
124C4⋊C4×C61—21no—
125C4⋊C4×C61—21no—
126C4⋊C4×C61—21no—
127C4⋊C4×C61—41no—
128C4⋊C4×C61—41no—
129C22⋊C4×C61—41no—
130C22⋊C4×C61—41no—
131C22⋊C4×C61—41no—
132C42×C61—41no—
133C42×C61C2×D411yescharacteristic
134C42×C61C2×D411yescharacteristic
135C42×C61#4111yescharacteristic
136C42×C61C2411yescharacteristic
137#11758201—44no—
138#11758199—44no—
139#11758198—22no—
140#11758198—22no—
141C4⋊Dic61—21no—
142C4⋊Dic61C2×D412yes—
143C2×#92418—22no—
144C2×#92418—22no—
145C2×#92418—22no—
146C8×D61—82no—
147C8×D61—82no—
148C23×C61—41no—
149C23×C61#12111yescharacteristic
150C22×D61—22no—
151C22×D61—24no—
152C22×D61—81no—
153C22×D61—81no—
154Q8×C61—42no—
155Q8×C61—81no—
156Q8×C61—81no—
157D4×C61—42no—
158D4×C61—42no—
159D4×C61—81no—
160D4×C61—82no—
161C2×C4×C61—22no—
162C2×C4×C61—22no—
163C2×C4×C61—41no—
164C2×C4×C61—41no—
165C2×C4×C61—41no—
166C2×C4×C61—41no—
167C2×C4×C61—82no—
168C2×C4×C61#12111yescharacteristic
169C2×C4×C61#12111yescharacteristic
170C2×C4×C61#12111yescharacteristic
171C2×C4×C61C22×D411yescharacteristic
172C2×C4×C61C22×D411yescharacteristic
173C2×C4×C61C2×#4111yescharacteristic
174C2×Dic61—22no—
175C2×Dic61—81no—
176D244—24no—
177D244—22no—
178D244#13514yes—
179C4×D61—42no—
180C4×D61—42no—
181C4×D61—82no—
182C8×C61—22no—
183C8×C61—22no—
184#92418—42no—
185#92418—42no—
186#92418—42no—
187#19394—611nomaximal, Sylow
188C22×C61—21no—
189C22×C61—21no—
190C22×C61—21no—
191C22×C61—81no—
192C22×C61—81no—
193C22×C61—81no—
194C22×C61C2×#12111yescharacteristic
195C2×D61—24no—
196C2×D61—42no—
197C2×D61—42no—
198C2×D61—84no—
199C4×C61—21no—
200C4×C61—21no—
201C4×C61—21no—
202C4×C61—21no—
203C4×C61—81no—
204C4×C61—81no—
205C4×C61—81no—
206C4×C61C2×#13512yes—
207C4×C61#57712yes—
208Dic61—42no—
209#3228D6111yescharacteristic, complemented
210#3109—611no—
211#3101—611no—
212#2972—611no—
213#1681—612no—
214#1677—612no—
215#1675—612no—
216#1671—612no—
217#1572—611no—
218#1554—611no—
219#1543—611no—
220C2×C61—41no—
221C2×C61—161no—
222C2×C61#304211yescharacteristic
223C2×C61#305011yescharacteristic
224C2×C61#306811yescharacteristic
225D61—44no—
226#533C2×D6111yescharacteristic
227#532C2×D6111yescharacteristic
228C2×C4×D4—611no—
229#500C2×D6111yescharacteristic, complemented
230#498C2×D6111yescharacteristic, complemented
231#486C2×D6111yescharacteristic, complemented
232#486C2×D6111yescharacteristic, complemented
233#483—1221no—
234#482—22no—
235#481—1221no—
236#478—22no—
237#476—612no—
238#476—612no—
239#461—612no—
240#459—612no—
241#450—1221no—
242#447—1221no—
243#442—1222no—
244C4×SD16—1222no—
245C8×D4—612no—
246C8×D4—612no—
247C8×D4—1222no—
248#432—611no—
249#431—611no—
250#431—611no—
251#431C2×D6111yescharacteristic
252C2×C22⋊C8—1222no—
253#335—612no—
254#334—612no—
255#333—612no—
256#333—612no—
257#332—612no—
258#331—612no—
259C61#1939411yescharacteristic, complemented, Sylow
260C2×#41—21no—
261C22×D4—611no—
262C23×C4—1221no—
263C2×SD16—1222no—
264C2×D8—1222no—
265C22×C8—1222no—
266C22×C8—1222no—
267C4⋊Q8C22×D6111yescharacteristic
268C4⋊Q8C22×D6111yescharacteristic
269C4⋊Q8C22×D6111yescharacteristic
270#127C22×D6111yescharacteristic
271#125—21no—
272#124—21no—
273C22⋊Q8—21no—
274C22⋊Q8—21no—
275#121—21no—
276#121—21no—
277C4×Q8—21no—
278C4×Q8—21no—
279C4×D4—21no—
280C4×D4—21no—
281C4×D4—612no—
282C4×D4—612no—
283C4×D4—1221no—
284C4×D4—1221no—
285#117—21no—
286C2×C4⋊C4—611no—
287C2×C22⋊C4—1221no—
288C2×C42C22×D6111yescharacteristic
289#107—22no—
290#106—22no—
291#106—1222no—
292#106—1222no—
293C4⋊C8—612no—
294C4⋊C8—612no—
295C4⋊C8—612no—
296C4⋊C8C22⋊D6112yes—
297#103—1222no—
298#102—22no—
299#102—22no—
300#102—1222no—
301#102—1222no—
302#102—1222no—
303C22⋊C8—1221no—
304C22⋊C8—1221no—
305C22⋊C8—1221no—
306C22⋊C8—1222no—
307C22⋊C8—1222no—
308C22⋊C8C22⋊D6112yes—
309#97—1221no—
310C4×C8—1221no—
311C4×C8—1221no—
312C4×C8C22×D6111yescharacteristic
313C4×C8C22×D6111yescharacteristic
314C24—1221no—
315#41—82no—
316C2×Q8—21no—
317C2×Q8—21no—
318C2×Q8—21no—
319C2×D4—21no—
320C2×D4—21no—
321C2×D4—21no—
322C2×D4—41no—
323C2×D4—612no—
324C2×D4—1224no—
325C2×D4—1221no—
326C22×C4—41no—
327C22×C4—1221no—
328C22×C4—1222no—
329C22×C4—1222no—
330C22×C4—4882no—
331C22×C4D4×D6111yescharacteristic
332C22×C4#1175822411yescharacteristic
333C22×C4C2×C22⋊D6111yescharacteristic
334SD16—2444no—
335D8—2444no—
336C2×C8—1222no—
337C2×C8—1222no—
338C2×C8—1222no—
339C2×C8—4882no—
340C2×C8—4882no—
341C2×C8C2×C22⋊D6112yes—
342C2×C8C2×C22⋊D6112yes—
343C4⋊C4—21no—
344C4⋊C4—21no—
345C4⋊C4—21no—
346C4⋊C4—21no—
347C4⋊C4—21no—
348C4⋊C4—21no—
349C4⋊C4—41no—
350C4⋊C4—41no—
351C4⋊C4—612no—
352C4⋊C4—1221no—
353C22⋊C4—41no—
354C22⋊C4—41no—
355C22⋊C4—41no—
356C22⋊C4—1222no—
357C22⋊C4—1222no—
358C42—41no—
359C42D4×D6111yescharacteristic
360C42#1175822411yescharacteristic
361C42C2×C22⋊D6111yescharacteristic
362C42C23×D6111yescharacteristic
363C23—41no—
364C23—1224no—
365C23—1222no—
366C23—4881no—
367C23—4881no—
368C23#1451974811yescharacteristic
369Q8—42no—
370Q8—81no—
371Q8—81no—
372D4—42no—
373D4—42no—
374D4—81no—
375D4—82no—
376D4—614no—
377D4—1224no—
378D4—1222no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—82no—
386C2×C4—1222no—
387C2×C4—4882no—
388C2×C4—4881no—
389C2×C4—2442no—
390C2×C4—2442no—
391C2×C4#1451974811yescharacteristic
392C2×C4#1451974811yescharacteristic
393C2×C4#1451974811yescharacteristic
394C2×C4C2×D4×D6111yescharacteristic
395C2×C4C22×C22⋊D6111yescharacteristic
396C2×C4C2×#1175822411yescharacteristic
397C8—22no—
398C8—22no—
399C8—2442no—
400C8—2442no—
401C8—2442no—
402C22—21no—
403C22—21no—
404C22—21no—
405C22—81no—
406C22—81no—
407C22—81no—
408C22—1224no—
409C22—4884no—
410C22—2442no—
411C22—2442no—
412C22C2×#1451974811yescharacteristic
413C4—21no—
414C4—21no—
415C4—21no—
416C4—21no—
417C4—81no—
418C4—81no—
419C4—81no—
420C4—2442no—
421C4#135×D6112yes—
422C4#1684057012yes—
423C2—41no—
424C2—161no—
425C2—2444no—
426C2#2145532711yescharacteristic
427C2#2145539611yescharacteristic
428C2#2145554311yescharacteristic
subgroupG/H
centerC22C2×#14519748
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3228C2
socleC22×C61C2×#121
prime radical#21455906C2
hypercenter#3228D61
hypocenterC61#19394
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3228

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #21455823C2semidirect
3. #21454842C2semidirect
4. #21445253C2semidirect
5. #21445225C2semidirect
6. #21445215C2semidirect
7. #21445191C2semidirect
8. #21444599C2semidirect
9. #21444529C2semidirect
10. #21444481C2semidirect
11. C61×#3228C2semidirect
30. #16839515C22semidirect
31. #16839513C22semidirect
32. #16839511C22semidirect
33. #16839511C22semidirect
34. #16839508C22semidirect
35. #16839506C22semidirect
38. C61×#500C22semidirect
39. C61×#498C22semidirect
40. C61×#486C22semidirect
41. C61×#486C22semidirect
209. #3228D61semidirect
229. #500C2×D61semidirect
230. #498C2×D61semidirect
231. #486C2×D61semidirect
232. #486C2×D61semidirect
259. C61#19394semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#214559062
#214558232
#214548422
#214452532
#214452252
#214452152
#214451912
#214445992
#214445292
#214444812
C61×#32282
#1939461